Using the Surface Force Apparatus (SFA) technique, direct force-distance measurements between mica surfaces immersed in a polymer-doped lamellar phase are presented. In the aqueous layers separating the charged and undulating bilayers of sodium dodecyl sulfate (SDS)/octanol, water is replaced by a polyethylene glycol aqueous solution (PEG, M-w similar to 20000 g/mol). As the mesophases align with the layers parallel to the solid mica surfaces the measurements yield structural information on the confined mesophases as well as bulk elastic properties. Keeping the membrane volume fraction constant the PEG dissolution results in a marked decrease in the elastic compressibility modulus, (B) over bar. This indicates a softening of the repulsive interactions between the lamellae mediated by polymer adsorption on the bilayers/aqueous solution interface. Insights into the behavior of the mesophases in confined geometries and thereby their stabilization mechanism are gained by noting a plastic contribution due to the presence of edge dislocations in the wedgelike confinement geometry.